<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">A</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="seq_4.html#V2">bounded_below</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> , <font color="Maroon">B</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="seq_4.html#V2">bounded_below</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> ;<br/>

<b>set </b><font color="Maroon">r</font> =  <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">B</font></span>)</span>;<br/>
<a NAME="E1:64"/><i><font color="Green">E34</font></i>: 
(  <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">B</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font> &amp;  <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">B</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">B</font> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T17">XXREAL_0:17</a></i>;<br/>
<a NAME="E2:64"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">t</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font> holds <br/><font color="Olive">t</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">B</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">t</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:64_1"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font>
 ;<br/>

<a NAME="E2:64_1"/><b>then </b>
( <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> or <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E3:64_1"/><b>then </b>
(  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t</font> or  <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">B</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t</font> )
 
<b>by </b><i><a class="ref" href="seq_4.html#D5">SEQ_4:def 5</a></i>;<br/>
<b>hence </b><a NAME="E4:64_1"/>
 <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">B</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:64"/>
for <font color="Olive">r1</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st ( for <font color="Olive">t</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font> holds <br/><font color="Olive">t</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">r1</font> ) holds <br/> <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">B</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">r1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">r1</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:64_2"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">t</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font> holds <br/><font color="Olive">t</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r1</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">t</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>assume </b><a NAME="E1:64_2_1"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/><a NAME="E2:64_2_1"/><b>then </b>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E3:64_2_1"/>
<font color="Maroon">t</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r1</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E3:64_2"/><b>then </b><i><font color="Green">E45</font></i>: 
 <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r1</font>
 
<b>by </b><i><a class="ref" href="seq_4.html#T60">SEQ_4:60</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">t</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>assume </b><a NAME="E1:64_2_2"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/><a NAME="E2:64_2_2"/><b>then </b>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E3:64_2_2"/>
<font color="Maroon">t</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r1</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E5:64_2"/><b>then </b>
 <a href="pscomp_1.html#K3">inf</a> <font color="Maroon">B</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r1</font>
 
<b>by </b><i><a class="ref" href="seq_4.html#T60">SEQ_4:60</a></i>;<br/>
<b>hence </b><a NAME="E6:64_2"/>
 <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">B</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r1</font>
 <b>by </b><i><a class="ref" href="sprect_1.html#T5" target="_self">Th5</a>, <a class="ref" href="square_1.html#T39">SQUARE_1:39</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:64"/>
 <a href="pscomp_1.html#K3">inf</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xxreal_0.html#K3">min</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K3">inf</a> <font color="Maroon">B</font></span>)</span>
 <b>by </b><i><a class="ref" href="sprect_1.html#T4" target="_self">Th4</a>, <a class="ref" href="seq_4.html#T61">SEQ_4:61</a></i>;<br/>


</div>
